Theorem trsspwALT 39045

Description: Virtual deduction proof of the left-to-right implication of dftr4 4757. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4757 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
trsspwALT	|- ( Tr A -> A C_ ~P A )

Proof of Theorem trsspwALT

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3591	. . 3 |- ( A C_ ~P A <-> A. x ( x e. A -> x e. ~P A ) )
2		idn1 38790	. . . . . . 7 |- (. Tr A ->. Tr A ).
3		idn2 38838	. . . . . . 7 |- (. Tr A ,. x e. A ->. x e. A ).
4		trss 4761	. . . . . . 7 |- ( Tr A -> ( x e. A -> x C_ A ) )
5	2, 3, 4	e12 38951	. . . . . 6 |- (. Tr A ,. x e. A ->. x C_ A ).
6		vex 3203	. . . . . . 7 |- x e. _V
7	6	elpw 4164	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
8	5, 7	e2bir 38858	. . . . 5 |- (. Tr A ,. x e. A ->. x e. ~P A ).
9	8	in2 38830	. . . 4 |- (. Tr A ->. ( x e. A -> x e. ~P A ) ).
10	9	gen11 38841	. . 3 |- (. Tr A ->. A. x ( x e. A -> x e. ~P A ) ).
11		biimpr 210	. . 3 |- ( ( A C_ ~P A <-> A. x ( x e. A -> x e. ~P A ) ) -> ( A. x ( x e. A -> x e. ~P A ) -> A C_ ~P A ) )
12	1, 10, 11	e01 38916	. 2 |- (. Tr A ->. A C_ ~P A ).
13	12	in1 38787	1 |- ( Tr A -> A C_ ~P A )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-tr 4753  df-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)